If all moments are known then can be computed using series expansions for exponentials but this seems to be a difficult problem: the fourth moments of GARCH were obtained by effort here: fourth-moment-garch-he-terasvirta-1997.

While closed form expression for kurtosis of these models are difficult problems, it is quite easy to compute the kurtosis by Monte Carlo simulations, which tell us that the excess kurtosis of the model over 1899 stocks is 17.8 +/- 1.4 which is thus producing leptokurtic returns (albeit not with the wide distribution of the empirical kurtosis of returns). Here is the code and the results for the Monte Carlo simulations.